#403090

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,15)(7,17)(8,21)(9,23,25,30)(10,14,26,16)(12,19)(13,29,31,24)(18,27)(20,28)(22,32)(33,36)(34,35);(5,7)(6,12,15,21)(8,10)(9,18)(11,20)(13,27,31,35)(14,17,16,28)(19,26)(22,24)(23,36,30,32)(25,34)(29,33);(5,8,10,20,11,19,26,7)(6,12,14,17,15,21,16,28)(9,22,24,34,25,33,29,18)(13,27,30,32,31,35,23,36);(2,4)(5,9,10,24,11,25,26,29)(6,13,16,23,15,31,14,30)(7,18,8,22,20,34,19,33)(12,27,28,36,21,35,17,32)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310286);